score:6
The LOC expressly prohibits obtaining a library card as a souvenir. So the strictest answer to your question is: You can't.
However, it's fairly easy to fudge, if you're either dishonest and say your intended purpose is to do research. Or, my preferred method: Actually spend some time doing some research! I spent an hour or two researching George Washington while I was there, which was an interesting experience (I learned a lot more about the LOC than I did about Mr. Washington).
The actual registration process is fairly straight forward (details here). I believe it took me about half an hour, and I had to fill out a few forms (I believe one said I would be using the card to perform actual research, and not as a souvenir).
At least that's how it worked for me about 3 years ago when I was there.
Upvote:3
I can't speak to the Library of Congress but from experience, the British Library has a similar system - you can preregister online or turn up in person, in which case you fill out more or less the same registration form on a terminal in the reader registration office before going through the same interview and approval process. (Worth noting in passing - the BL's interview checks you do need to use the collections, so 'get a souvenir card' probably wouldn't work there. The LoC seems to be less restrictive.)
In the BL's case, they encourage preregistration as this lets the reader order material before getting admitted - otherwise you turn up, register, and then have to wait while material is delivered to the reading room, which can take a while. I have dug around on the LoC's site for a while but can't seem to find out whether a pre-registration allows you to order material in advance. If so, this would be the major advantage.
If not... well, as the comments note, it'll save five minutes and avoid any nasty surprises when you don't have the right information to hand. (Every librarian at a major institution has stories about a researcher who flies half-way across the world to consult a vital manuscript, gets there, and discovers they need one particular piece of ID that they didn't think to bring...)